Topology of non-negatively curved manifolds

TL;DR

This paper classifies manifolds with the same cohomology as Eschenburg spaces, showing many are diffeomorphic and admit non-negative curvature or Einstein metrics, expanding understanding of non-negatively curved manifolds.

Contribution

It classifies certain cohomology ring types of manifolds and identifies diffeomorphisms with Eschenburg spaces, also establishing curvature properties.

Findings

Many manifolds are diffeomorphic to Eschenburg spaces.

Total spaces admit non-negative sectional curvature.

Some admit Einstein metrics.

Abstract

We examine several classes of manifolds which have the same cohomology ring as an Eschenburg space (a family of biquotients which is a main source of manifolds with positive curvature). One family are the 3-sphere bundles over CP^2. Another are the circle bundles over a base, which itself is one of the family of CP^1 bundles over CP^2. We classify such manifolds up to diffeomorphism using the Kreck-Stolz invariants. Comparisons of the invariants is then used to find many diffeomorphism of the total space of these bundles with positively curved Eschenburg spaces. The total space of each bundle (in the case where the bundle is not spin) also admit a metric with non-negative sectional curvature, and in some cases an Einstein metric as well.